Problem: $ A = \left[\begin{array}{rr}1 & 2 \\ 3 & 4\end{array}\right]$ $ F = \left[\begin{array}{rrr}-2 & 3 & 3 \\ -2 & 3 & 0\end{array}\right]$ What is $ A F$ ?
Answer: Because $ A$ has dimensions $(2\times2)$ and $ F$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ A F = \left[\begin{array}{rr}{1} & {2} \\ {3} & {4}\end{array}\right] \left[\begin{array}{rrr}{-2} & \color{#DF0030}{3} & \color{#9D38BD}{3} \\ {-2} & \color{#DF0030}{3} & \color{#9D38BD}{0}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{1}\cdot{-2}+{2}\cdot{-2} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{-2}+{2}\cdot{-2} & ? & ? \\ {3}\cdot{-2}+{4}\cdot{-2} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{-2}+{2}\cdot{-2} & {1}\cdot\color{#DF0030}{3}+{2}\cdot\color{#DF0030}{3} & ? \\ {3}\cdot{-2}+{4}\cdot{-2} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{1}\cdot{-2}+{2}\cdot{-2} & {1}\cdot\color{#DF0030}{3}+{2}\cdot\color{#DF0030}{3} & {1}\cdot\color{#9D38BD}{3}+{2}\cdot\color{#9D38BD}{0} \\ {3}\cdot{-2}+{4}\cdot{-2} & {3}\cdot\color{#DF0030}{3}+{4}\cdot\color{#DF0030}{3} & {3}\cdot\color{#9D38BD}{3}+{4}\cdot\color{#9D38BD}{0}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-6 & 9 & 3 \\ -14 & 21 & 9\end{array}\right] $